Windowed-Sinc Filters

Windowed-sinc filters are used to separate one band of frequencies from another. They are very stable, produce few surprises, and can be pushed to incredible performance levels. These exceptional frequency domain characteristics are obtained at the expe

16

h [i ] ' sin (2BfCi )

i B

Windowed-sinc filters are used to separate one band of frequencies from another They are very stable, produce few surprises, and can be pushed to incredible performance levels These exceptional frequency domain characteristics are obtained at the expense of poor performance in the time domain, including excessive ripple and overshoot in the step response When carried out

by standard convolution, windowed-sinc filters are easy to program, but slow to execute Chapter

18 shows how the FFT can be used to dramatically improve the computational speed of these filters

Strategy of the Windowed-Sinc

Figure 16-1 illustrates the idea behind the windowed-sinc filter In (a), the

frequency response of the ideal low-pass filter is shown All frequencies below

the cutoff frequency, f C, are passed with unity amplitude, while all higher frequencies are blocked The passband is perfectly flat, the attenuation in the stopband is infinite, and the transition between the two is infinitesimally small

Taking the Inverse Fourier Transform of this ideal frequency response produces the ideal filter kernel (impulse response) shown in (b) As previously discussed (see Chapter 11, Eq 11-4), this curve is of the general form: sin(x)/x, called

the sinc function, given by:

Convolving an input signal with this filter kernel provides a perfect low-pass

filter The problem is, the sinc function continues to both negative and positive infinity without dropping to zero amplitude While this infinite length is not

a problem for mathematics, it is a show stopper for computers.

w[i ] ' 0.54 & 0.46 cos (2 Bi /M )

EQUATION 16-1

The Hamming window These

windows run from i ' 0 to M,

for a total of M % 1 points.

w[i ] ' 0.42 & 0.5 cos (2 Bi /M ) % 0.08 cos(4Bi /M )

EQUATION 16-2

The Blackman window.

FIGURE 16-1 (facing page)

Derivation of the windowed-sinc filter kernel The frequency response of the ideal low-pass filter is shown

in (a), with the corresponding filter kernel in (b), a sinc function Since the sinc is infinitely long, it must be truncated to be used in a computer, as shown in (c) However, this truncation results in undesirable changes

in the frequency response, (d) The solution is to multiply the truncated-sinc with a smooth window, (e), resulting in the windowed-sinc filter kernel, (f) The frequency response of the windowed-sinc, (g), is smooth and well behaved These figures are not to scale.

To get around this problem, we will make two modifications to the sinc function in (b), resulting in the waveform shown in (c) First, it is truncated

to M % 1 points, symmetrically chosen around the main lobe, where M is an

even number All samples outside these M % 1 points are set to zero, or simply ignored Second, the entire sequence is shifted to the right so that it runs from

0 to M This allows the filter kernel to be represented using only positive indexes While many programming languages allow negative indexes, they are

a nuisance to use The sole effect of this M / 2 shift in the filter kernel is to shift the output signal by the same amount

Since the modified filter kernel is only an approximation to the ideal filter kernel, it will not have an ideal frequency response To find the frequency response that is obtained, the Fourier transform can be taken of the signal in (c), resulting in the curve in (d) It's a mess! There is excessive ripple in the passband and poor attenuation in the stopband (recall the Gibbs effect discussed in Chapter 11) These problems result from the abrupt discontinuity

at the ends of the truncated sinc function Increasing the length of the filter kernel does not reduce these problems; the discontinuity is significant no matter

how long M is made.

Fortunately, there is a simple method of improving this situation Figure (e)

shows a smoothly tapered curve called a Blackman window Multiplying the truncated-sinc, (c), by the Blackman window, (e), results in the

windowed-sinc filter kernel shown in (f) The idea is to reduce the abruptness of the

truncated ends and thereby improve the frequency response Figure (g) shows this improvement The passband is now flat, and the stopband attenuation is

so good it cannot be seen in this graph

Several different windows are available, most of them named after their

original developers in the 1950s Only two are worth using, the Hamming

window and the Blackman window These are given by:

Figure 16-2a shows the shape of these two windows for M ' 50 (i.e., 51 total points in the curves) Which of these two windows should you use? It's a trade-off between parameters As shown in Fig 16-2b, the Hamming

window has about a 20% faster roll-off than the Blackman However,

Time Domain

Frequency

-0.5 0.0 0.5 1.0 1.5

fc

a Ideal frequency response

Sample number

-0.5

0.0

0.5

1.0

1.5

b Ideal filter kernel

Frequency

-0.5 0.0 0.5 1.0 1.5

fc

d Truncated-sinc frequency response

Sample number

-0.5

0.0

0.5

1.0

1.5

M abrupt end

c Truncated-sinc filter kernel

Sample number

-0.5

0.0

0.5

1.0

1.5

M

e Blackman or Hamming window

Frequency Domain

Frequency

-0.5 0.0 0.5 1.0

1.5

g Windowed-sinc frequency response

fc

Sample number

-0.5

0.0

0.5

1.0

1.5

M

f Windowed-sinc filter kernel

FIGURE 16-1

Amplitude Amplitude

Sample number

0.0

0.5

1.0

1.5

a Blackman and Hamming window

Blackman Hamming

Frequency

0.0 0.5 1.0

1.5

b Frequency response

Hamming Blackman

Frequency

-120 -100 -80 -60 -40 -20 0 20

40

c Frequency response (dB)

Blackman

Hamming

FIGURE 16-2

Characteristics of the Blackman and Hamming

windows The shapes of these two windows are

shown in (a), and given by Eqs 16-1 and 16-2 As

shown in (b), the Hamming window results in about

20% faster roll-off than the Blackman window.

However, the Blackman window has better

stop-band attenuation (Blackman: 0.02%, Hamming:

0.2%), and a lower passband ripple (Blackman:

0.02% Hamming: 0.2%). Amplitude (dB)

(c) shows that the Blackman has a better stopband attenuation To be exact,

the stopband attenuation for the Blackman is -74dB (-0.02%), while the Hamming is only -53dB (-0.2%) Although it cannot be seen in these graphs,

the Blackman has a passband ripple of only about 0.02%, while the Hamming

is typically 0.2% In general, the Blackman should be your first choice; a slow roll-off is easier to handle than poor stopband attenuation

There are other windows you might hear about, although they fall short of the

Blackman and Hamming The Bartlett window is a triangle, using straight lines for the taper The Hanning window, also called the raised cosine

window, is given by: w[i] ' 0.5 & 0.5 cos(2 Bi /M) These two windows have about the same roll-off speed as the Hamming, but worse stopband attenuation (Bartlett: -25dB or 5.6%, Hanning -44dB or 0.63%) You might also hear of

a rectangular window This is the same as no window, just a truncation of

the tails (such as in Fig 16-1c) While the roll-off is -2.5 times faster than the Blackman, the stopband attenuation is only -21dB (8.9%)

Designing the Filter

To design a windowed-sinc, two parameters must be selected: the cutoff frequency, f , and the length of the filter kernel, M The cutoff frequency

0.0 0.5 1.0 1.5

b Roll-off vs cutoff frequency

FIGURE 16-3

Filter length vs roll-off of the windowed-sinc filter As shown in (a), for M = 20, 40, and 200, the transition bandwidths are BW = 0.2, 0.1, and 0.02 of the sampling rate, respectively As shown in (b), the shape of the frequency response does not change with different cutoff frequencies In (b), M = 60.

Frequency

0.0

0.5

1.0

1.5

M=20

a Roll-off vs kernel length

BW

EQUATION 16-3

Filter length vs roll-off The length of the

filter kernel, M, determines the transition

bandwidth of the filter, BW This is only an

approximation since roll-off depends on the

particular window being used

is expressed as a fraction of the sampling rate, and therefore must be between

0 and 0.5 The value for M sets the roll-off according to the approximation:

where BW is the width of the transition band, measured from where the curve

just barely leaves one, to where it almost reaches zero (say, 99% to 1% of the curve) The transition bandwidth is also expressed as a fraction of the sampling frequency, and must between 0 and 0.5 Figure 16-3a shows an example of how this approximation is used The three curves shown are generated from filter kernels with: M ' 20, 40, and 200 From Eq 16-3, the transition bandwidths are: BW ' 0.2, 0.1, and 0.02, respectively Figure (b) shows that the shape of the frequency response does not depend on the cutoff frequency selected

Since the time required for a convolution is proportional to the length of the

signals, Eq 16-3 expresses a trade-off between computation time (depends on the value of M) and filter sharpness (the value of BW) For instance, the 20%

slower roll-off of the Blackman window (as compared with the Hamming) can

be compensated for by using a filter kernel 20% longer In other words, it could be said that the Blackman window is 20% slower to execute that an equivalent roll-off Hamming window This is important because the execution speed of windowed-sinc filters is already terribly slow

As also shown in Fig 16-3b, the cutoff frequency of the windowed-sinc filter

is measured at the one-half amplitude point Why use 0.5 instead of the

h [i ] ' K sin (2BfC(i & M /2))

i & M /2 0.42 & 0.5 cos

2Bi

M % 0.08 cos 4Bi

M

EQUATION 16-4

The windowed-sinc filter kernel The cutoff frequency, f C, is expressed as a

fraction of the sampling rate, a value between 0 and 0.5 The length of the filter

kernel is determined by M, which must be an even integer The sample number

i, is an integer that runs from 0 to M, resulting in M%1 total points in the filter

kernel The constant, K, is chosen to provide unity gain at zero frequency To

avoid a divide-by-zero error, for i ' M/2, use h[i ] ' 2 Bf C K.

standard 0.707 (-3dB) used in analog electronics and other digital filters? This

is because the windowed-sinc's frequency response is symmetrical between the

passband and the stopband For instance, the Hamming window results in a

passband ripple of 0.2%, and an identical stopband attenuation (i.e., ripple in

the stopband) of 0.2% Other filters do not show this symmetry, and therefore have no advantage in using the one-half amplitude point to mark the cutoff frequency As shown later in this chapter, this symmetry makes the

windowed-sinc ideal for spectral inversion.

After f C and M have been selected, the filter kernel is calculated from the

relation:

Don't be intimidated by this equation! Based on the previous discussion, you

should be able to identify three components: the sinc function, the M/2 shift, and the Blackman window For the filter to have unity gain at DC, the constant

K must be chosen such that the sum of all the samples is equal to one In

practice, ignore K during the calculation of the filter kernel, and then normalize

all of the samples as needed The program listed in Table 16-1 shows how this

is done Also notice how the calculation is handled at the center of the sinc,

, which involves a division by zero

i ' M/2

This equation may be long, but it is easy to use; simply type it into your computer program and forget it Let the computer handle the calculations If you find yourself trying to evaluate this equation by hand, you are doing something very very wrong

Let's be specific about where the filter kernel described by Eq 16-4 is located

in your computer array As an example, M will be chosen to be 100 Remember, M must be an even number The first point in the filter kernel is

in array location 0, while the last point is in array location 100 This means that the entire signal is 101 points long The center of symmetry is at point 50, i.e., M/2 The 50 points to the left of point 50 are symmetrical with the 50 points to the right Point 0 is the same value as point 100, and point 49 is the same as point 51 If you must have a specific number of samples in the filter kernel, such as to use the FFT, simply add zeros to one end or the other For example, with M ' 100, you could make samples 101 through 127 equal to zero, resulting in a filter kernel 128 points long

Filter kernel

Sample number

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

1.2

b f C = 0.015

M = 500

Sample number

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

a f C = 0.015

M = 500

Sample number

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

1.2

d f C = 0.04

M = 500

Sample number

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

c f C = 0.04

M = 500

Sample number

-0.02

0.00

0.02

0.04

0.06

0.08

0.10

150

e f C = 0.04

M = 150

Step response

Sample number

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

1.2

f f C = 0.04

M = 150

FIGURE 16-4

Example filter kernels and the corresponding step responses The frequency of the sinusoidal oscillation is approximately equal to the cutoff frequency, f C , while M determines the kernel length.

Figure 16-4 shows examples of windowed-sinc filter kernels, and their corresponding step responses The samples at the beginning and end of the filter kernels are so small that they can't even be seen in the graphs Don't make the mistake of thinking they are unimportant! These samples may

be small in value; however, they collectively have a large effect on the

FIGURE 16-5

Example of windowed-sinc filters The alpha and beta rhythms in an EEG are separated by low-pass and high-pass filters with M ' 100 The program to implement the low-pass filter is shown in Table 16-1 The program

for the high-pass filter is identical, except for a spectral inversion of the low-pass filter kernel

Frequency

0.0

0.5

1.0

1.5

a Low-pass filter

Frequency

0.0 0.5 1.0

1.5

b High-pass filter

Amplitude Amplitude

Alpha wave Beta wave Alpha wave Beta wave

performance of the filter This is also why floating point representation is typically used to implement windowed-sinc filters Integers usually don't have enough dynamic range to capture the large variation of values contained in the filter kernel How does the windowed-sinc filter perform in the time domain?

Terrible! The step response has overshoot and ringing; this is not a filter for

signals with information encoded in the time domain

Examples of Windowed-Sinc Filters

An electroencephalogram, or EEG, is a measurement of the electrical activity of the brain It can be detected as millivolt level signals appearing

on electrodes attached to the surface of the head Each nerve cell in the brain generates small electrical pulses The EEG is the combined result of

an enormous number of these electrical pulses being generated in a (hopefully) coordinated manner Although the relationship between thought and this electrical coordination is very poorly understood, different frequencies in the EEG can be identified with specific mental states If you close your eyes and relax, the predominant EEG pattern will be a slow oscillation between about 7 and 12 hertz This waveform is called the

alpha rhythm, and is associated with contentment and a decreased level of

attention Opening your eyes and looking around causes the EEG to change

to the beta rhythm, occurring between about 17 and 20 hertz Other

frequencies and waveforms are seen in children, different depths of sleep, and various brain disorders such as epilepsy

In this example, we will assume that the EEG signal has been amplified by analog electronics, and then digitized at a sampling rate of 100 samples per second Acquiring data for 50 seconds produces a signal of 5,000 points Our goal is to separate the alpha from the beta rhythms To do this, we will design

a digital low-pass filter with a cutoff frequency of 14 hertz, or 0.14

Frequency (discrete)

0.00

0.50

1.00

1.50

a Frequency response

Frequency (hertz)

-120 -100 -80 -60 -40 -20 0 20

40

b Frequency response (dB)

FIGURE 16-6

Example of a windowed-sinc band-pass filter This filter was designed for a sampling rate of 10 kHz When referenced to the analog signal, the center frequency of the passband is at 2 kHz, the passband is 80 hertz, and the transition bands are 50 hertz The windowed-sinc uses 801 points in the filter kernel to achieve this roll-off, and a Blackman window for good stopband attenuation Figure (a) shows the resulting frequency response on a linear scale, while (b) shows it in decibels The frequency axis in (a) is expressed as a fraction of the sampling frequency, while (b) is expressed in terms of the analog signal before digitization.

of the sampling rate The transition bandwidth will be set at 4 hertz, or 0.04 of the sampling rate From Eq 16-3, the filter kernel needs to be about 101 points long, and we will arbitrarily choose to use a Hamming window The program in Table 16-1 shows how the filter is carried out The frequency response of the filter, obtained by taking the Fourier Transform of the filter kernel, is shown in Fig 16-5

In a second example, we will design a band-pass filter to isolate a signaling

tone in an audio signal, such as when a button on a telephone is pressed We

will assume that the signal has been digitized at 10 kHz, and the goal is to isolate an 80 hertz band of frequencies centered on 2 kHz In terms of the sampling rate, we want to block all frequencies below 0.196 and above 0.204 (corresponding to 1960 hertz and 2040 hertz, respectively) To achieve a transition bandwidth of 50 hertz (0.005 of the sampling rate), we will make the filter kernel 801 points long, and use a Blackman window Table 16-2 contains

a program for calculating the filter kernel, while Fig 16-6 shows the frequency

response The design involves several steps First, two low-pass filters are

designed, one with a cutoff at 0.196, and the other with a cutoff at 0.204 This

second filter is then spectrally inverted, making it a high-pass filter (see

Chapter 14, Fig 14-6) Next, the two filter kernels are added, resulting in a

band-reject filter (see Fig 14-8) Finally, another spectral inversion makes

this into the desired band-pass filter

Pushing it to the Limit

The windowed-sinc filter can be pushed to incredible performance levels

without nasty surprises For instance, suppose you need to isolate a 1 millivolt signal riding on a 120 volt power line The low-pass filter will need

100 'LOW-PASS WINDOWED-SINC FILTER

110 'This program filters 5000 samples with a 101 point windowed-sinc filter,

120 'resulting in 4900 samples of filtered data.

130 '

140 DIM X[4999] 'X[ ] holds the input signal

150 DIM Y[4999] 'Y[ ] holds the output signal

160 DIM H[100] 'H[ ] holds the filter kernel

170 '

180 PI = 3.14159265

190 FC = 14 'Set the cutoff frequency (between 0 and 0.5)

200 M% = 100 'Set filter length (101 points)

210 '

220 GOSUB XXXX 'Mythical subroutine to load X[ ]

230 '

240 ' 'Calculate the low-pass filter kernel via Eq 16-4

250 FOR I% = 0 TO 100

260 IF (I%-M%/2) = 0 THEN H[I%] = 2*PI*FC

270 IF (I%-M%/2) <> 0 THEN H[I%] = SIN(2*PI*FC * (I%-M%/2)) / (I%-M%/2)

280 H[I%] = H[I%] * (0.54 - 0.46*COS(2*PI*I%/M%) )

290 NEXT I%

300 '

310 SUM = 0 'Normalize the low-pass filter kernel for

320 FOR I% = 0 TO 100 'unity gain at DC

330 SUM = SUM + H[I%]

340 NEXT I%

350 '

360 FOR I% = 0 TO 100

370 H[I%] = H[I%] / SUM

380 NEXT I%

390 '

400 FOR J% = 100 TO 4999 'Convolve the input signal & filter kernel

410 Y[J%] = 0

420 FOR I% = 0 TO 100

430 Y[J%] = Y[J%] + X[J%-I%] * H[I%]

440 NEXT I%

450 NEXT J%

460 '

470 END

TABLE 16-1

a stopband attenuation of at least -120dB (one part in one-million for those that refuse to learn decibels) As previously shown, the Blackman window only provides -74dB (one part in five-thousand) Fortunately, greater stopband attenuation is easy to obtain The input signal can be filtered using a conventional windowed-sinc filter kernel, providing an intermediate signal The intermediate signal can then be passed through the filter a second time, further increasing the stopband attenuation to -148dB (1 part

in 30 million, wow!) It is also possible to combine the two stages into a

single filter The kernel of the combined filter is equal to the convolution of

the filter kernels of the two stages This also means that convolving any

filter kernel with itself results in a filter kernel with a much improved

stopband attenuation The price you pay is a longer filter kernel and a slower roll-off Figure 16-7a shows the frequency response of a 201 point low-pass filter, formed by convolving a 101 point Blackman windowed-sinc with itself Amazing performance! (If you really need more than -100dB of stopband attenuation, you should use double precision Single precision